Non radical intervals with musical significance

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Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.

There are many non-radical intervals which have musical significance. By non-radical is meant a number that cannot be written in the form [math](a/b)^c[/math], where [math]a[/math] and [math]b[/math] are integers and [math]c[/math] is rational. What follows is a list of musically significant non-radical intervals.

Ratio Cents Name(s) Musical significance
[math]2^{1/\phi} \approx 1.5348[/math] 741.641 Logarithmic phi Divides the octave into two parts, one being phi times larger than the other in cents.
[math]\phi = \dfrac{\sqrt{5}+1}{2} \approx 1.6180[/math] 833.090 Acoustic phi
Golden ratio
Linear phi
"Linear phi," the unique interval whose continued fraction approximations converge more slowly than any other number. This is due to the continued fraction representation only containing 1's, as well as a general consequence of Dirichlet's approximation theorem.
[math]e \approx 2.7183[/math] 1731.23 Natave
In Gene's black magic formulas, it is mathematically more "natural" to consider the number of divisions to the "e-tave" rather than the octave.
[math]e^{2\pi} \approx 535.4917[/math] 10877.66 The zeta function has units that are given as divisions of the interval [math]e^{2\pi}[/math].